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Fractal Cities is the pioneering study of the development and use of fractal geometry for understanding and planning the physical form of cities, showing how this geometry enables cities to be simulated throughcomputer graphics. The book explains how the structure of cities evolve in ways which at first sight may appear irregular, but when understood in terms of fractals reveal a complex and diverse underlying order. The book includes numerous illustrations and 16 pages full-color plates of stunning computer graphics, along with explanations of how to construct them. The authors provide an accessible and thought-provoking introduction to fractal geometry, as well as an exciting visual understanding of the formof cities. This approach, bolstered by new insights into the complexity of social systems, provides one of the best introductions to fractal geometry available for non-mathematicians and social scientists.
Fractal Cities is useful as a textbook for courses on geographic information systems, urban geography, regional science, and fractal geometry. Planners and architects will find that many aspects of fractal geometry covered in this book are relevant to their own interests. Those involved in fractals and chaos, computer graphics, and systems theory will also find important methods and examples germane to their work.
Michael Batty is Director of the National Center for Geographic Information and analysis in the State University of New York at Buffalo, and has worked in planning theory and urban modeling. Paul Longley is a lecturer in geography at the University of Bristol, and is involved in the development of geographic information systems in urban policy analysis.

Richly illustrated, including 16 pages of full-color plates of brilliant computer graphics
Provides an introduction to fractal geometry for the non-mathematician and social scientist
Explains the influence of fractals on the evolution of the physical form of cities ... Read more

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This book covers analysis on fractals, a developing area of mathematics that focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals. ... Read more

Customer Reviews (1)

Metrics, fractals, analysis...
Fractals make headlines from time to time[--are they everywhere?], and and they make beautiful color pictures; but they are also part of a substantial mathematical theory, even one with an exciting mathematical history. This lovely book presents the subject elegantly in a way that it can be taught to students. It starts with the basics, then systematically, step by step, it builds up the central results. Great for the classrom, or for selfstudy! In view of the many applications to geometric analysis, to PDE, and to statistics, it is likely that fractal geometry will soon be a standard math course taught in many (more) math departments. Example: By now it is widely recognized that the selfsimilarity aspects of the wavelet algorithms are key to their sucess. Compared to other books at the same level, for example, by Falconer, 1990, and the equally attractive one from 1985[Falconer: The geometry of fractal sets], Kigami's book stresses the analysis of the Laplace operator. Falconer covers the theory more generally, and his books have a slightly more potential theoretic bent. ... Read more

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This volume contains the proceedings of the first IFIP conference on fractals. The mathematical concept of fractal was coined by B.B. Mandelbrot in the early seventies to characterize objects with self-similar structures at all levels. Now the field of fractals is being subject to intensive research and an extensive growth of applications in many sciences. This first IFIP conference, intended as a start of periodic conferences on fractals, enables a productive cross fertilization of ideas among specialists in fractals and in information processing. In addition to the volume's 44 colour illustrations, on page v is a rare view of fractals and the Portuguese architecture at the time of the navigators, produced by Master Lima de Freitas, a leading Portuguese artist and member of the conference organizing committee. The design illustrates the symbiosis of mathematics and human artifacts represented by the real world. ... Read more

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This text will be a valuable tool and reference work for all those interested in fractal forms and their properties. The past decade has seen an almost exponential amount of work done in the field of non-integral dimensions. Upon publication of Mandelbrot's Objets Fractals (1975), a great number of researchers understood the interest of fractals. Many tried, by means of the theory, to set up immediately useable tools, including the author, who opted for an experimental approach and developed instrumentation to explore the widest possible range in both time and space (time resolution of the laser spectroscopy of biomolecules). In this spirit, these expert researchers organized monthly ``Hausdorff seminars on the notion of non-integral dimension and its applications'' to tackle the problems confronting practical scientists. This work captures, with many illustrative examples, the main themes of each session. ... Read more

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Fractal analysis research is expanding into a variety of engineering domains. The strong potential of this work is now beginning to be seen in important applications in real industrial situations. Recent research progress has already led to new developments in domains such as signal processing and chemical engineering, and the major advances in fractal theory that underlie such developments are detailed here. New domains of applications are also presented, among them environmental science and rough surface analysis. Sections include multifractal analysis, iterated function systems, random processes, network traffic analysis, fractals and waves, image compression, and applications in physics. Fractals in Engineering emphasizes the connection between fractal analysis research and applications to industry. It is an important volume that illustrates the scientific and industrial value of this exciting field. ... Read more

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The nine review articles in this book give a fairly completesurvey of theapplications of fractals to astronomical andastrophysical problems. Thebook is self-contained andwritten with the particular aim of teachinggraduatestudents how to apply fractal geometry to new problems.After ageneral introduction to the basic concepts,applications are given to astrophysical dynamical systems,to strange attractors of stellar pulsations, to the solargranular pattern, and to both the large-scalestructure ofthe universe - a central theme of the book - and toastronomical data analysis. ... Read more

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Proceedings of the International Conference on Fractals held in Denver, U.S.A. Topics include: Crystal pattern formation, Fractal metrology, Fluid flow analysis, Estimation of multifractal measures and more. ... Read more

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Fractal geometry is revolutionizing the descriptive mathematics of applied materials systems. Rather than presenting a mathematical treatise, Brian Kaye demonstrates the power of fractal geometry in describing materials ranging from Swiss cheese to pyrolytic graphite. Written from a practical point of view, the author assiduously avoids the use of equations while introducing the reader to numerous interesting and challenging problems in subject areas ranging from geography to fine particle science. The second edition of this successful book provides up-to-date literature coverage of the use of fractal geometry in all areas of science.

From reviews of the first edition:

'...no stone is left unturned in the quest for applications of fractal geometry to fine particle problems....This book should provide hours of enjoyable reading to those wishing to become acquainted with the ideas of fractal geometry as applied to practical materials problems.' MRS Bulletin ... Read more

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Fractal geometry has become popular in the last 15 years,its applicationscan be found in technology, science, oreven arts. Fractal methods andformalism are seen today as ageneral, abstract, but nevertheless practical instrument forthe description of nature in a wide sense. But itwasComputer Graphics which made possible the increasingpopularity offractals several years ago, and long aftertheir mathematical formulation.The two disciplines aretightly linked.The book contains the scientificcontributions presentedin an international workshop in the "Computer GraphicsCenter" in Darmstadt, Germany. Thetarget of the workshop wasto present the wide spectrumof interrelationships and interactions between FractalGeometry and Computer Graphics. The topics vary fromfundamentals and new theoretical results to variousapplications and systems development. All contributions areoriginal, unpublished papers.The presentations have beendiscussed in two working groups; the discussion results,together with actual trends and topics of future research,are reported in the last section.The topics of the book are divides into foursections:Fundamentals, Computer Graphics and Optical Simulation,Simulation of Natural Phenomena, Image Processing and ImageAnalysis. ... Read more

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The fractal concept has become an important tool for understanding irregular complex systems in various scientific disciplines. This book discusses in great detail fractals in biology, heterogeneous chemistry, polymers, and the earth sciences. Beginning with a general introduction to fractal geometry it continues with eight chapters on self-organized criticality, rough surfaces and interfaces, random walks, chemical reactions, and fractals in chemisty, biology, and medicine. A special chapter entitled "Computer Exploration of Fractals, Chaos, and Cooperativity" presents computer demonstrations of fractal models. 14 executable programs are included on a 3 1/2" MS-DOS diskette and run on any IBM or compatible computer. ... Read more

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This volume contains 11 papers presented at the First IMA Conference on Fractal Geometry: Mathematical Methods, Algorithms and Applications, held at De Montfort University in September 2000. Researchers include many from Europe, Russia, and the United States, giving a multinational emphasis to the fractal geometry problems described. Emphasized are the mathematical exposure given to a problem and the practicalities required to create and implement an algorithm. ... Read more

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Based on the proceedings of a conference held at Newnham College, Cambridge in December 1990. ... Read more

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This volume brings together a number of distinctive contributions in the areas of fractals, chaos, and the interrelationship between the two domains. These contributions cover a wide variety of application areas, and this indicates the extent to which fractals and chaotic phenomena are being studied in the various disciplines. It is anticipated that the interdisciplinary nature of the subject will increase, which in turn will yield useful information on the potential (and also limitations in some cases) of fractals and chaos as modeling tools for the investigation of various natural and scientific phenomena. It is hoped that an understanding of fractals and chaos will lead to a common basis for examining the growth, development, organization, and behaviour of complex dynamical systems, many of which make up the natural world of which we are part. It is anticipated that the investigations of fractal structure associated with phase portraits will be an exciting area for future work. ... Read more

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This insightful work explains Mandelbrot's fractal geometry and describes some of its most interesting applications. Fractal geometry exploits a characteristic property of the real world--self-similarity--to find simple rules for the assembly of complex natural objects.Beginning with the foundations of measurement in Euclidean geometry, the authors progress from analogues in the geometry of random fractals to applications spanning the natural sciences, including the developmental biology of neurons and pancreatic islets, fluctuations of bird populations, patterns in vegetative ecosystems, and even earthquake models.Written to enable students and researchers to master the methods of this timely subject, the book steers a middle course between the formality of many papers in mathematics and the informality of picture-orientated books on fractals. It is both a logically developed text and an essential "fractals for users" handbook.It is an essential resource for researchers and students in ecology, biology, applied mathematics, and plant and environmental sciences. ... Read more

Customer Reviews (2)

For mathematicians, not for beginners
The rise of fractal processes is the result of success and failure. It has been known for decades that continuous models were inadequate to simulate many phenomena. To paraphrase Benoit Mandelbrot, the father of fractals, "clouds are not spheres, mountains are not cones and trees are not lollipops." However, there were very few alternatives. Some early work on structures now recognized as fractal was done decades ago, but technical factors limited the scope. It was necessary for inexpensive computers and graphical devices to arise before fractals could truly be explored. The confluence of these two forces has led to a flood of work on fractals, the consequences of which will not be known for many years to come. However, it is clear that some knowledge of fractals is rapidly becoming a requirement in many disciplines, from physics to economics.
The authors here eschew a popular treatment of fractals, relying on mathematical explanations with few pictures. One immediately realizes that it is a scholarly work when a quick scan reveals that there are no color plates. Applications are used throughout. Earthquakes, pancreatic islets, temperature variations, rainfall data and vegetative ecosystems are some of the topics examined using fractals. But be prepared to perform some mathematics if you wish to understand.
And that is the outstanding feature of this book. There are a growing number of popular mathematics books that contain sections on fractals. Unfortunately, it is always more of the same. A description of the generation of the Mandelbrot and Julia sets, with color pictures. This book is designed to teach the working scientist the techniques needed to apply fractal models to their specific discipline. The mathematics is hard, complete and applicable. Listings of the many programs used to generate fractals are at the end of the book and the language is Turbo Pascal.
If your goal is to learn fractals for scientific applications, then this is the book for you. Otherwise, your best choice is one of the many popular books that use verbal explanations.

Originally published in School Science and Mathematics, reprinted with permission.

not for the beginner
It's an odd book.

According to the book's review is in between the formality of many papers and a picture oriented book.

The first two lines of its table of contents are:

chapter. 1 Our view of nature page. 7...chapter. 2 Fractals and power law scaling 15...36

The chapter 1 is whatany picture intended book on fractals has. The chapter 2, where the funbegins and basic concepts as "power law", "fractaldimension" are introduced is incomprehensible.

Nevertheless it seemsto be an interesting book about applications of fractals. I'm talking about"Part III", page > 80 to 233 (the end).

It's curious andinconsistent book at the beginning: it talks to beginners in a way that isincomprehensible for a beginner!.

If you are a beginner in fractals as Iam, you need another book!. ... Read more

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Published in cooperation with the National Council of Teachers of Mathematics, this text is based on lectures given to various communities of teachers and students. Written especially for teachers and intended for classroom use at high school and college levels, the book also can be used for independent study. ... Read more

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Fractals for the Classroom
Actually, this is a pretty good book, but if you are seeking a simple way to understand and begin to understand Fractals, this book still require a good amount of math knowledge.I have yet to find a cogent, simple book for the curious, but not math minded that is not so "dumbed" down to the point of just showing pretty pictures. ... Read more

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Fractal geometry is the formal study of mathematical shapes that display a progression of never-ending, self-similar, meandering detail from large to small scales. It has the descriptive power to capture, explain, and enhance one's appreciation of and control over complex diversity. Natural shapes and rhythms, such as leaves, tree branching, mountain ridges, flood levels of a river, wave patterns, and nerve impulses, display this cascading behaviour. These fractal concepts are found in many fields, from physics to musical composition.
Architecture and design, concerned with control over rhythm, and with such fractal concepts as the progression of forms from a distant view down to the intimate details, can benefit from the use of this relatively new mathematical tool. Fractal geometry is a rare example of a technology that reaches into the core of design composition, allowing the architect or designer to express a complex understanding of nature.
The exposition of the book is at a level suitable for applied scientists, architects, and students with a modest background in mathematics. It is well illustrated and has numerous examples from which to learn the underlying concepts and their applications. Thus the book is addressed to a wide audience with a multiplicity of interests in new compositional ideas.

Customer Reviews (1)

Only book I've seen that understands both subjects
Bovill does a good job of explaining the principles of fractal geometry in layman's terms for those not already familiar with them. I was particularly impressed with his understanding of how fractal dimension varies with scaleand which scales are important for architecture. However, I wasdisappointed that there were no references made to structural properties offractals and how they might have excellent strength to weight ratios for awide range of scales. Everything was taken from a design perspectiveconcerned with texture. ... Read more

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From reviews of the first edition:

"In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."

Mathematics Teaching

"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."

Christoph Bandt, Mathematical Reviews

"...not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

Customer Reviews (4)

some new material->same approach
I bought the first edition of this in the early 90's
and was disappointed that it didn't have the Mandelbrot or other complex dynamics
in it. Dr. Edgar has updated the older book with Julias, multifractals
and Superfractals, but has stayed true to his topological measure theory
Hausdorff space approach. He never updates his Biscovitch-Ursell functions
to 2d and 3d parametrics or the unit Mandelbrot cartoon method.
Some of his definitions are still so minimal
that duplicating the fractals needs much more information?!
The text is still the good place to begin, but
it is a shame that Dr. Edgar has not kept up
with many of the developments in the field. Zipf and Per Bak
are left out, but my double V L-system made the index as a picture.

A nice book
I liked this book because it provided me with a new perspective on metricspaces, in using them as a basis learning about fractals. I think it servesas a nice book for an undergraduate to read and get enthused about studyingfractals at a higher level.

Good starting point to study fractal geometry.
This book could be used as a bridge between traditional books on topology-analysis and the speciallized treatises on fractal geometry. More a catalog of definitions, methods, and references than a course text, itcovers the fundamental topological and measure-theoretic concepts needed tounderstand the principles of some of the different dimension theories thatexist. But warning: the book is far away of being a complete exposition onany of the subjects it includes.

Suitable for 3rd-year undergrads.Interesting examples and exercises. Extensive bibliography.

Please checkmy other reviews in my member page (just click on my name above).

A difficult but worthy book!
The programs are in LOGO: don't let the turtles fool you, this is the real stuff by a master teacher. It is hard and the examples are even harder. The problem sets are at times impossible, but in the end Dr. Edgar delivers: understanding! Your unique Associates ID is:thefractaltransl. ... Read more

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Concepts, methods and techniques of statistical physics in the study of correlated, as well as uncorrelated, phenomena are being applied ever increasingly in the natural sciences, biology and economics in an attempt to understand and model the large variability and risks of phenomena. This is the first textbook written by a well-known expert that provides a modern up-to-date introduction for workers outside statistical physics. The emphasis of the book is on a clear understanding of concepts and methods, while it also provides the tools that can be of immediate use in applications. Although this book evolved out of a course for graduate students, it will be of great interest to researchers and engineers, as well as to post-docs in geophysics and meteorology. ... Read more

Customer Reviews (3)

very complete
As I am teaching the statistical mechanics part of a graduate class in mathematics,I was looking for a textbook on complex systems & statistical physicswith derivations, intuitions, and some physical examples. I did not realize that I was looking too far --Sornette, with hom I correspond regularly, is well known for his contributions and his prolific output (actually some physicists make fun of the quantity of papers he writes). So his book did not come to mind. I once stumbled on a problem with the derivation of preferential attachments;he recommended his book, which I took with a grain a salt. After spending some time working the derivations on scalable laws, extreme value theory, renormalization groups in this book, I elected to use it as my textbook. There is no equivalent. I have a dozen such yellow manuals; this one is complete and ultimately clearest.
I do not know of a better textbook.

One of the best textbooks ever written for graduate students
Didier Sornette who is potentialy nobelisable realises with this book a great present for everybody interested to the recent progresses toward the physics of critical phenomena.
Particularly, it shows the way that some seismologists follow toward the ultimate goal to predict the event of large earthquakes. If this task was impossible yesterday,Didier Sornette shows that it is now became realisable.

A must for understanding complexity
Sornette's book is quite an achievement both in quantity and
quality. The presentation remains informal and quite readable; it reads like a physics textbook, not a math textbook. The references are very extensive (a total of 832! altogether) and they are a very valuable component of the book. In fact much of the book is about the reference material. You might choose to read the book instead of the 832 references... I think this is
the point...

There's probably nothing wrong with this book besides the fact that it throws it all at you at a high degree of sophistication and in as terse a way as possible, it seems. It's a unique and beautiful achievement but because it is so dense with information and insight, it seems every word counts for ten and you'll want to read several chapters again and again. Also, even though there is a clear unifying theme from chapter to chapter, the book simply ends almost like in the middle of a sentence. After machinegunning out 392 pages of material at research level spanning quite a few scientific fields, there is absolutely no attempt to put it all together. It's up to you to do it and it almost seems like the author is indirectly suggesting you start reading it all over again to "get it"... So, for the second edition, perhaps the author will be bold and add ten pages of wrap up material at the end so that this will read less like an atlas. Apart from that, it's the best! ... Read more

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In this book we generate graphic images using the software Mathematica thus providing a gentle and enjoyable introduction to this rather technical software and its graphic capabilities.
The programs we use for generating these graphics are easily adaptable to many variations.

These graphic images are enhanced by introducing a variety of different coloring techniques.

Detailed instructions are given for the construction of some interesting 2D and 3D fractals using iterated functions systems as well as the construction of many different types of Julia sets and parameter sets such as the Mandelbrot set.

The mathematics underlying the theory of Iterated function systems and Julia sets is given an intuitive explanation, and references are provided for more detailed study.

· Brilliant Graphic images
· Interesting Introduction to Mathematica for Beginners
· Easy constructions
· A variety of Coloring Techniques
· Programs Easily Adaptable to Many Variations
· Constructions useful for Dynamics and Fractals Courses ... Read more